Dynamic analysis and optimal control of stochastic information cross-dissemination and variation model with random parametric perturbations

Information dissemination has a significant impact on social development. This paper considers that there are many stochastic factors in the social system, which will result in the phenomena of information cross-dissemination and variation. The dual-system stochastic susceptible-infectious-mutant-recovered model of information cross-dissemination and variation is derived from this problem. Afterward, the existence of the global positive solution is demonstrated, sufficient conditions for the disappearance of information and its stationary distribution are calculated, and the optimal control strategy for the stochastic model is proposed. The numerical simulation supports the results of the theoretical analysis and is compared to the parameter variation of the deterministic model. The results demonstrate that cross-dissemination of information can result in information variation and diffusion. Meanwhile, white noise has a positive effect on information dissemination, which can be improved by adjusting the perturbation parameters.

Reviewer #3: 1. Response to comment: (In abstract the author claims that white noise has a positive effect on information dissemination, which can be improved by adjusting the perturbation parameters.They may provide the evidence for this either theoretically or numerically.In the current version I did not see any evidence for this claim.) Response: Thank you very much for your comments!Your suggestions are of great help to the improvement of the paper.In the original manuscript, our expression is not comprehensive and careful enough.We would like to make a further explanation on this point.According to your request, we have further explained how white noise effect on information dissemination in numerical simulations in the revised manuscript (Page 20, line 413 -420).The details are as follows: "In Figure 5 and Figure 7, the fluctuating lines represent the population density changes after adding random disturbances.And the stable lines represent the population density changes without adding random disturbances.It can be seen that the population densities with added random perturbations are higher than that without added random perturbations.From Figures 5 (c "In social systems multiple pieces of information sometimes coexist.The phenomenon of coexistence and intersection in the process of multiple information dissemination is similar to the spread of viruses in biology.For example, when the SARS-CoV-2 virus spread in 2020, the presence of its original strain did not cause the extinction of other strains in the body.Instead, there were multiple strains coexisting, and even new mutated strains were produced through cross transmission, such as Omicron (B.1.1.529).The phenomenon of coexisting and producing mutated strains of this virus can be analogized and applied to the spread of multiple information.Multiple similar information may have coexisting relationships, and after prolonged cross propagation, the content expressed by each information may deviate from the original information, similar to the phenomenon of multiple viruses coexisting and producing mutated viruses.This paper proposes a stochastic susceptible-infectious-mutantrecovered model that considers information cross-dissemination and variation, and then demonstrates the existence of global positive solutions.After calculating the sufficient conditions for information disappearance and information stationary distribution, the appropriate parameters are selected as control variables.The numerical simulation validates the rationality of the proposed method is finally validated through numerical 5. Response to comment: (From page 7 onwards there is too many mathematical equations.The author may delete the unnecessary mathematical equations.) Response: Thank you very much for the attentive and earnest comments!Your suggestions will help improve the reading experience of the manuscript.Because we want to fully reflect the work of this manuscript as much as possible in the article.
Therefore, the necessary steps were included in the main text.Your reminder can help us learn simpler steps to prove in future research.We will continue to study the papers you have provided us and delve into the proof methods from these excellent papers."Due to the range of the parameters has not been explicitly given in previous studies.
Therefore, the values of the parameters in the model can be given according to the conditions given by Theorem 3.1 and Theorem 4.1."8. Response to comment: (Which parameter is more sensitive and why in numerical simulations section.The author must write some sentences about this.) Response: Thanks very much for your comments.We do regret for the lack of explanation of parameters sensitivity due to the omission of necessary instructions in the original manuscript, which has brought you a dissatisfactory reading experience.
According to your comments and suggestions, we have added some sentences to explanation of parameters sensitivity in the revised manuscript (Page 20, line 421 -430).The details are as follows: "Next, in order to verify the effectiveness of the optimal control strategy, and then observe the change of densities of  !() and  " () when the optimal control strategy is adopted.Here, the optimal control is adopted for the control variables  !! ,  !" ,  "" ,  "! and other parameters remain unchanged.Figure 9 and Figure 10 confirm the densities of  !() and  " () change over time when  ( ( = 1~8) = 0.001 and  ( ( = 1~8) = 0.0001 under constant control measure and optimal control.From Figure 9 and Figure 10, it can be seen that adopting optimal control for control parameter  !! ,  !" ,  "" ,  "! can further enhance information dissemination.
That is to say, the cross-contact rate is more sensitive to the dissemination of information."9. Response to comment: (Author may look for some punctuation, typos and editing issues.) Response: Thanks very much for your comments.Your comments are conducive us to be more careful.It is more helpful for us to improve our good scientific research literacy.
According to your request, we have revised upon all punctuation, typos and editing issues once again.And we have already put them right in the revised version.As these minor errors are more, we have not marked each and every of them corresponding page and line numbers in details. (3) The information dissemination may be effectively facilitated by controlling the perturbation parameters; unlike earlier research, the optimal control strategy provided in this paper is based on the optimal value calculated by the control variables.
To sum up, for positive information, it is necessary to give full play to the activity of the social system itself, and to introduce a large number of stochastic components into the social system to improve the information dissemination.For negative information, on the other hand, it is crucial to eliminate the external environment's uncertain factors and reduce the impact of uncontrollable factors on the social system in order to inhibit its dissemination.In future study, the role of stochastic environmental factors on information dissemination in social systems will be further investigated, and a v process-driven information dissemination model will be developed."

Special thanks to you for your good comments!
)-(d) and Figure 7 (c)-(d), the white noise disturbance enhances the population densities of information dissemination group  ! and  " .From this, it can be seen that white noise disturbance has a positive effect on information dissemination." 2. Response to comment: (The introduction should also make a compelling case for why the study is useful along with a clear statement of its novelty or originality by providing relevant information and providing answers to basic questions such as: i.What is already known in the literature?ii.What was done and how it was done?) Response: Thank you very much for your incisive comments and thorough reminders.Your comments are extremely important for improving the practical application of the model established in this paper.According to your comments and suggestions, we have added a clear statement of its novelty or originality in the revised manuscript (Page 3, line108 -Page 4, line 124).The details are as follows: to comment: (The author may also add some recants work about the current study related to their work in introduction part.They may add the following but not mandatory.https://doi.org/10.1038/s41598-023-41861-4https://doi.org/10.1063/1.5016680https://doi.org/10.1016/j.aej.2023.01.027 doi: 10.3934/math.2023210)Response: Thanks for your comments and suggestions.We have downloaded and read these references.These references are closely related to the research of this paper, and they deepen our understanding.It is more helpful for us to clearly understand the the application of stochastic differential equations.Therefore, we have added refers to these useful recent papers in the revised manuscript (Page 2, line 60 -69).The details are as follows: "Chu et al.(2023)[30] constructed an  #  $  %  malnutrition model with random perturbations and crossover effects, and using fractional differential equations analysis deterministic-stochastic model.Rashid et al.(2023)[31] constructed an  &  '  '&  &  '  '&  of the co-infection of the fractional pneumonia and typhoid fever disease stochastic model with cost-effective techniques and crossover effects.Ali et al.(2023)[32] gave the dynamics analysis and simulations of stochastic COVID-19 epidemic model using Legendre spectral collocation method.In addition, Khan et al.(2018)[33] studied on the application of Legendre spectral-collocation method to delay differential and stochastic delay differential equation."4. Response to comment: (Even though all the figures are available at the end, but there is no figure in the text and only captions of the figures are there, please see page from 3 onwards.)Response: Thank you very much for your careful reading, helpful comments and constructive suggestions, which has significantly improved the presentation of our manuscript.In the original manuscript, we did not put the figures in the text.For this, we are very sorry for the inconvenience caused to your reading.According to your comments and suggestions, we have inserted all the figures into the main text in the revised manuscript (Figure 1 to 10).

6.
Response to comment: (Improve the quality of the figures (from figure 4 and onwards).)Response: Thank you very much for your incisive comments and thorough reminders.We are very sorry that our image resolution was indeed very low, resulting in the information displayed being not clear enough.According to your comments and suggestions, we have further improved the resolution of all the figures in the revised manuscript (Figure 1 to 10). 7. Response to comment: (In numerical simulation section, what is the criteria to choose the values of the parameters involve in model equation (1) and equation (5).)Response: Thank you very much for the attentive and earnest comments!We do regret for the lack of smooth logical narration due to the omission of necessary instructions in the original manuscript, which has brought you a dissatisfactory reading experience.According to your comments and suggestions, we have added instructions for the criteria to choose the values of the parameters involve in model equation (1) and equation (5) in the revised manuscript (Page 18, line 381 -Page 19, line 384).The details are as follows: 10. Response to comment: (The conclusion section is too long.In general conclusion consist of what is claimed is achieved.)Response: Thanks for your sincere comments and reminders, which are indeed to the point.We agree that the conclusion of this article is indeed too long.According to your comments and suggestions, we have reduced the textual description in the conclusion section and highlighted the key points more in the revised manuscript (Page 21, line 442 -460).The details as following: "This study investigates the influence of multi-population information crossdissemination, mutation, and white noise disturbance on information dissemination.And developed the stochastic model.The following results are achieved through examination of this paper: (1) White noise disturbance can facilitate the information dissemination, and stochastic environmental factors play a positive role in information dissemination.(2) As disturbance intensity increases, the stochasticity of the model gradually enhances, and the fluctuation of information dissemination trend becomes more apparent.